Homogenisation of Temperature and Precipitation Time Series with ACMANT3: Method Description and Efficiency Tests

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INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 37: 1910–1921 (2017) Published online 19 July 2016 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/joc.4822

Homogenisation of temperature and precipitation time series with ACMANT3: method description and efficiency tests

P. Domonkosa* andJ.Collb a Centre for Climate Change, University of Rovira i Virgili, Tortosa, Spain b Irish Climate Analysis and Research UnitS (ICARUS), Department of Geography, Maynooth University, Co Kildare, Ireland

ABSTRACT: The development of Adapted Caussinus–Mestre Algorithm for homogenising Networks of Temperature series (ACMANT), one of the most successful homogenisation methods tested by the European project COST ES0601 (HOME) has been continued. The third generation of the software package ‘ACMANT3’ contains six programmes for homogenising temperature values or precipitation totals. These incorporate two models of the annual cycle of temperature biases and homogenisation either on a monthly or daily time scale. All ACMANT3 programmes are fully automatic and the method is therefore suitable for homogenising large datasets. This paper describes the theoretical background of ACMANT and the recent developments, which extend the capabilities, and hence, the application of the method. The most important novelties in ACMANT3 are: the ensemble pre-homogenisation with the exclusion of one potential reference composite in each ensemble member; the use of ordinary kriging for weighting reference composites; the assessment of seasonal cycle of temperature biases in case of irregular-shaped seasonal cycles. ACMANT3 also allows for homogenisation on the daily scale including for break timing assessment, gap filling and ANOVA application on the daily time scale. Examples of efficiency testsof monthly temperature homogenisation using artificially developed but realistic test datasets are presented. ACMANT3 canbe characterized by improved efficiency in comparison with earlier ACMANT versions, high missing data tolerance and improved user friendliness. Discussion concerning when the use of an automatic homogenisation method is recommended is included, and some caveats in relation to how and when ACMANT3 should be applied are provided.

KEY WORDS time series; homogenisation; ACMANT; spatial interpolation; surface air temperature; precipitation Received 12 February 2016; Revised 31 May 2016; Accepted 8 June 2016

1. Introduction observational and data transfer errors. Certain data quality problems can be eliminated by general quality con- For the analysis of climate change and climate variabil- trol (Durre et al., 2010; Menne et al., 2012), or with the ity the accuracy of observational data is of key importance analysis of the documents of the history of observations (Williams et al., 2012; Acquaotta and Fratianni, 2014). (so-called metadata, Bergstrom and Moberg, 2002; Pro- Although national meteorological services led by World hom et al., 2016). However, the statistical homogenisa- Meteorological Organisation (WMO) initiatives foster the tion of data provides additional quality control and allows production of high quality and comparable climatic data for improved temporal and spatial comparisons between temporally and spatially, technical changes in the mea- data for scientific purposes (Peterson et al., 1998; Beaulieu surement setup or observational practices often influence et al., 2008; Venema et al., 2012). When observational net- the usability of climate data records. Observational data works are sufficiently dense, relative homogenisation (i.e. can only be considered temporally homogeneous (here- homogenisation methods including spatial comparisons after: homogeneous) if temporal variations are exclusively of time series) can help to remove even relatively small influenced by weather and climate. In practice, several non-climatic biases from the data. Therefore, the statistical factors corrupt the homogeneity of climatic time series, methodology underpinning time series homogenisation is these include: station relocations, changes of instrumen- a widely studied topic of climatology (e.g. Series of Data tation, instrument position, site changes around the instru- Quality Control and Time Series Homogenisation, World ment, changes of the timing of reading instruments, etc. Meteorological Organisation, 1996–2014). (Aguilar et al., 2003; Menne et al., 2009; Acquaotta et al., Both the most common and frequent form of inhomo- 2016). A general observation is that long observational cli- geneity in a climate time series is the sudden shift of the matic records are seldom homogeneous, and that the qual- means, commonly referred to as a break or change-point. ity of climatic records may also be affected by occasional A set of breaks can be searched and corrected one-by-one in hierarchic structures (e.g. Alexandersson and Moberg, * Correspondence to: P. Domonkos, Centre for Climate Change, Uni- 1997), or jointly with appropriate mathematical tools. versity of Rovira i Virgili, Av. Remolins 13-15, 43500 Tortosa, Spain. When time series include multiple breaks, joint treatments E-mail: [email protected] have theoretical advantages over hierarchic techniques

(Szentimrey et al., 2014; Lindau and Venema, 2016), as Caussinus–Mestre Algorithm for homogenising Networks in hierarchic techniques early phase errors are delivered to of Temperature series, Domonkos, 2011b) and the other the later steps of the homogenisation process. For purposes is Homogenization software in R (HOMER, Mestre et al., of this study ‘multiple break method’ means a method 2013), the interactive homogenisation method officially with joint detection of inhomogeneities, and one which recommended by HOME. Both HOMER and ACMANT incorporates the joint calculation of correction terms provide additional functionality relative to the parent for adjusting inhomogeneities. As observed temperature method PRODIGE. Therefore recently, HOMER and time series include five to seven breaks per 100 years on ACMANT have been applied more frequently than average (Menne et al., 2009; Venema et al., 2012), or even PRODIGE. more due to hidden short-term biases (Domonkos, 2011a; After the termination of HOME, the development of Rienzner and Gandolfi, 2011) multiple break methods ACMANT has continued, and the second generation of are of key importance in providing high level solutions ACMANT (ACMANT2) already incorporated methods for homogenisation tasks particularly in relation to the for the precipitation homogenisation and for the treat- homogenisation of temperature. Efficiency tests prove ment of daily data through downscaling the monthly that multiple break methods generally outperform other homogenisation results to daily scale (Domonkos, 2014, homogenisation methods (Domonkos, 2011a,; Domonkos 2015a). Most recently the development of ACMANT3 has et al., 2011; Venema et al., 2012). Although some inho- improved the efficiency and user friendliness further, and mogeneities result in gradually increasing biases instead some of these improvements are detailed below. of abrupt shifts of the means (e.g. urbanisation), this effect ACMANT3 is a complex software package incorporat- has little impact on the rank order of method efficiencies ing six programmes, these are: temperature homogeni- (Domonkos, 2011a). sation with a sinusoid annual cycle of biases; temper- The organisation of this paper is as follows: The devel- ature homogenisation with an irregular annual cycle of opment of multiple break methods and particularly the biases; precipitation homogenisation. Each of the pre- development of Adapted Caussinus–Mestre Algorithm for ceding three has monthly and daily homogenisation ver- homogenising Networks of Temperature (ACMANT) is sions (http://www.c3.urv.cat/data.html); and in total the described in the next section; in Section 3, the novel fea- six programmes incorporate 174 sub-routines. The soft- tures of ACMANT3 compared with earlier ACMANT ver- ware package also includes auxiliary files to support net- sions are presented; some efficiency results are presented work construction. However, despite its complicated struc- in Section 4; and the paper ends with a discussion and some ture, ACMANT provides the fastest method implemen- recommendations in Section 5. tation among all the available automatic homogenisation methods. As both HOMER and ACMANT have been developed 2. Development of multiple break methods and from PRODIGE, the two new multiple break methods ACMANT have several similarities. Table 1 summarizes the main similarities and differences of the two methods. Note that Although statistical break detections and corrections although HOMER is only for monthly homogenisation, the have been studied and applied for at least 90 years joint use of HOMER and Spline Daily Homogenization (Conrad, 1925), the theory and development of multi- (SPLIDHOM, Mestre et al., 2011) can be applied for daily ple break homogenisation appeared only in the 1990s data homogenisation (www.homogenisation.org), and in coincident with the more widespread use of personal Table 1 it is considered the daily homogenisation version computers. At that time two approaches to multiple of HOMER. break homogenisation were established, namely Mul- tiple Analysis of Series for Homogenisation (MASH), Szentimrey, 1996, 1999) and PRODIGE (Caussinus 3. Methodological novelties of ACMANT3 and Mestre, 1996, 2004). These two methods differ markedly from each other: MASH uses multiple reference The full scientific description of ACMANT2 has been series, selects the set of breaks with hypothesis testing, published (Domonkos, 2014, 2015a), and therefore, only derives adjustments terms from the confidence intervals the new features of ACMANT3 in comparison with the belonging to the hypothesis test results, and the MASH earlier ACMANT versions are presented here. Appendix A approach to the final solution is iterative. By contrast, details the mathematical formulations of some steps of the PRODIGE uses pairwise comparisons, optimal step homogenisation connected to the content of this section. function fitting for break detection and the minimisation Note that some less important details are not shown due to of residual variance (ANOVA) for the adjustments of the complexity of the methodology. inhomogeneities in a procedure without iteration. Both To aid interpretation of the present description of the MASH and PRODIGE were among the most successful methodology, it should be recalled that in ACMANT, the methods tested by the European project COST ES0601 candidate series for homogenisation is compared with the (known as ‘HOME’, 2007–2011). During HOME, weighted average of other available series in the same two new multiple break methods were created based on climatic area. This weighted average series is referred to PRODIGE: one is the fully automatic ACMANT (Adapted as composite reference series and the contributor series as

Table 1. Similarities and differences between HOMER and routines employed by ACMANT3 are completely different ACMANT. and their core is the search for and application of ensemble minimums as adjustment terms. The underpinning concept HOMER ACMANT3 is that the application of false adjustments can be excluded Temperature homogenisation X X with high certainty by the use of ensemble minimums. Precipitation XX In the first phase, an ensemble of pre-homogenisation homogenisation is produced, and this always excludes one potential refer- Monthly homogenisation X X Daily homogenisation X X ence component from the homogenization. The main steps Pairwise comparison X of the pre-homogenisation are (1) creating relative time Composite reference series X series; (2) break detection on an annual scale and; (3) cal- Optimal step function fitting X X culation of adjustment terms with ANOVA (Caussinus and Derivatives of the C–L XXMestre, 2004; Domonkos, 2014) are performed for each criterion ensemble member. The adjustment terms indicated by the Network-wide joint X ensemble members are stored but not applied in this phase. segmentation Bivariate detection for XXIn the second phase, the minimum absolute value of the temperature (when stored adjustment terms is calculated for each time series applicable) and each year, and then, it is applied for obtaining the Bivariate detection for X pre-homogenisation result. If the signs of the adjustment precipitation (when terms of ensemble members are mixed for a particular year applicable) of a particular time series, then the result adjustment term Varied use of time resolution X is zero. See also Appendix A1.4. in break detection If univariate detection is applied, then the monthly Ensemble X pre-homogenisation adjustment terms of pre-homogenisation are constant Correction with ANOVA XXwithin a particular year, while the shape of the annual cycle model of adjustment terms is predefined when bivariate detection Iterations X is applied (Domonkos, 2014). The pre-homogenisation Monthly precision of breaks X X is performed twice in any programme of ACMANT3. Daily precision of large X In the first execution the cp coefficient of the modified magnitude breaks Caussinus–Lyazrhi (C–L) criterion (Appendix A1.3, Adjusting relative to the last XX homogeneous section formulas (8) and (9)) is elevated with 40%. The purpose Quantile dependent daily X of this distinction in parameter cp according to the stage correction terms of ACMANT3 procedure is to focus on the elimination of High missing data tolerance Xa Xa large inhomogeneities in the first pre-homogenisation. Completion of missing data X X Completion of data before X break detection 3.2. Ordinary kriging for determining the weights of Filtering of spatial outliers X X reference components Metadata use supported X Ordinary kriging is a widely applied tool for the produc- Graphical results X tion of optimally interpolated values of a meteorological Automatic execution X variable to a given location. Theoretically, ordinary kriging aThe missing data tolerance of ACMANT is higher than that of HOMER provides the optimal weights of reference series compos- (results of authors’ experiments, not shown). ites (Szentimrey, 2010), as the purpose of building composite reference series is to have another series beyond the reference composites, respectively. The difference (ratio) candidate series (i.e. the reference series) whose climatic of the candidate and reference series is referred to as variability is the same as that of the candidate series. How- relative time series in the homogenisation of temperature ever, in practice, the unavoidable inaccuracy of the large (precipitation) series. number of parameter estimates incorporated in ordinary kriging has the potential to reduce the efficiency. The more 3.1. Ensemble pre-homogenisation traditional way is applying the squared spatial correlations The aim of doing pre-homogenisation prior to the of increment time series as weights of the reference com- main homogenisation is to remove relatively large biases posites (Peterson and Easterling, 1994). before the main homogenisation. This excludes the pos- If the number of reference composites is N, then the sibility of potentially large biases in the reference com- number of estimated parameters is N in the traditional posites during the main homogenisation affecting the method, while for ordinary kriging it is 0.5 × N × (N − 1) accuracy of the final homogenisation results. The con- as ordinary kriging uses the entire covariance matrix of cept for ACMANT1 and ACMANT2 was that during the the candidate series and reference composites. According pre-homogenisation of the reference series of the later to tests (not shown) ordinary kriging performs better than candidate series, the later candidate series was excluded the traditional weighting of reference composites if it is from that pre-homogenisation. The pre-homogenisation applied with some restrictions: (1) Ordinary kriging is

© 2016 Royal Meteorological Society Int. J. Climatol. 37: 1910–1921 (2017) HOMOGENISATION OF TIME SERIES WITH ACMANT3 1913 applied only when N ≥ 6; (2) when an estimated weight platform-shaped inhomogeneity exists, or the daily values is negative, the applied weight is 0, as a negative weight are very scattered with significant mean bias. For identi- for a reference composite has no physical interpretation; fying the former case (which is likely the more frequent), (3) when an estimated weight would be higher than 0.4, a step function with two steps is fitted to the daily data the weights of the other reference composites are slightly of the relative time series spanning the period that starts elevated in order to reduce the overly strong influence 4 months before the outlier and ends 4 months after the of the most highly correlated reference composite. It is outlier (‘wide window’). The two breaks are searched applied to reduce the likelihood of importing any potential across a narrower window, specified for 1 month before inhomogeneity from the most closely correlated reference and after the outlier period with a minimum platform composite. length of 10 days, and univariate detection is applied. No platform-shaped inhomogeneity is identified when (1) 3.3. Irregular annual cycle of biases for temperature there are less than 4 months with observed monthly data inhomogeneities of the candidate series among the 8 months before and In the tropics and monsoon regions, the annual cycle of after the narrow window but within the wide window; or irradiation markedly differs from the semi-sinusoid cycle (2) the standard deviation of the relative time series values of the mid- and high-latitude regions. Therefore, the annual within the candidate platform-shaped inhomogeneity is at cycle of temperature biases is not sinusoid everywhere. In least twice as high as for all the values within the wide addition, biases of daily temperature minima are affected window. If a platform-shaped bias cannot be identified, more by the frequency of clear sky and calm weather then all the daily data of the outlier period are treated as conditions than by the annual cycle of irradiation. In missing data. ACMANT2, the model annual cycle of biases was constant when the sinusoid model was not applicable. ACMANT3 3.6. ANOVA on daily scale provides season-dependent adjustments for these cases, in In the late phase of daily homogenisation, the break times spite of the generally lower signal to noise ratio in the esti- are provided at daily resolution and ANOVA is then mates for seasonal than annual characteristics. The assess- applied to the daily resolution data to provide direct daily ment of the seasonal cycle of biases includes three main adjustment terms. The equation system for the practical steps: (1) Optimal step function fitting (Appendix A1.3) solution of the minimisation of the residual variance with is applied to the annual series of monthly temperatures ANOVA (Domonkos, 2014, 2015a) can be applied on any for each calendar month, with coarser time resolution than time scale. in the default application, this to allow for the lower signal to noise ratio for monthly compared to annual values. 3.7. Selective exclusion of years with too few observed Here, the minimum distance between two adjacent breaks data from the homogenisation is 5 years. (2) ANOVA is applied to the annual series of monthly values. In this step, the same ensemble procedure ACMANT’s default operational mode is to first infill is included for each month as in the pre-homogenisation data gaps with interpolated values, and all the routines (Appendix A1.4). (3) Finally, monthly estimates of adjust- of the software use continuous time series. However, ment terms derived from the ensemble calculations are when the number of synchronously available observed smoothed (Appendix A1.6). data in the network is very low, the reliability of interpolated values is correspondingly low, and hence, can 3.4. Daily precision of large-size breaks affect the efficiency of homogenisation. In ACMANT3, A step function with exactly one break is fitted to daily data years with too few observed values are excluded from of relative time series in a section, which includes both the most steps of the homogenisation procedure and are preceding and subsequent 6 months of the pre-estimated only included in the final interpolations for completing timing of the break. Univariate detection is applied, and time series and for applying final adjustments to correct the break is accepted in a narrower window only, which inhomogeneities. includes the preceding and subsequent 2 months of the Typically, years with intact observed data from less pre-estimated timing. This operation is done only when the than three stations are excluded. A yearly observation pre-estimated break magnitude is not lower than the 75% is classified as intact when the year has at least nine of the empirical standard deviation of daily data, this also observed monthly values (at least 9 months with at least considers either the entire window (1 year) or the narrower 75% complete observed daily values) in the case of window (4 months). When this operation is omitted for a monthly (daily) input data. Whereas in the case of pre- break of relatively small magnitude, the default timing of cipitation, only one missing daily data value excludes the the last calendar day of the pre-estimated month of the entire month from contributing positively to the evalu- break is retained. ation of the intact or non-intact character of the yearly observation. 3.5. Treatment of daily data when monthly outliers are detected 3.8. Elevated missing data tolerance A monthly outlier or an outlier value of the mean of a few Time series must comprise at least ∼10 years observed months period may indicate that on daily scale either a data, more precisely 114 monthly observations and a series

© 2016 Royal Meteorological Society Int. J. Climatol. 37: 1910–1921 (2017) 1914 P. DOMONKOS AND J. COLL may have more than a 100-year data gap. (See more Table 2. Summary statistics for the proportion of the test time requirements for data input in the ACMANT3 Manual, series completeness and the extent of missing data. Completeness http://www.c3.urv.cat/data.html.) and exterior missing data are proportional to the entire period As relative time series often have different lengths examined (100 years), while interior missing data to the periods (Domonkos, 2011b, 2014), long data gaps would result of observations. All values are in percentage units. in some reference series including virtual composites, i.e. Completeness Exterior Interior composites without their own observational data. In the missing missing present parameterisation scheme of ACMANT3, a ref- data data erence composite can be included if it has at least 12 Dataset (A) 89.0 10.0 1.1 observed monthly values falling within the corresponding Dataset (B) 50.1 39.2 17.7 period of the target relative series. This threshold is low, since experience with ACMANT3 shows that the inclusion of a very few additional observed values compensates well for the uncertainty associated with the inclusion of inter- are known. In good test datasets, both the climatic and polated values. inhomogeneity characteristics should be realistic and the dataset should be of sufficient size to make confident sta- 3.9. Modifications in the interpolation of monthly tistical estimations (Domonkos, 2013). In this section, values for substituting missing data the efficiencies of the earlier and new ACMANT versions are compared for the homogenisation of monthly The interpolation for a missing value of a candidate series temperatures with regular annual cycle of biases using is provided by the weighted average of the anomalies two test datasets. The homogeneous dataset was taken in other series of the network (‘partner series’) applied from the HOME benchmark, its large, 200 network sized to the missing data points, and these are then adjusted version is used (Venema et al., 2012). For the exercise to the climatic mean of the candidate series (Appendix here, climatic trends, additional noise and various kinds A1.5). Two changes are incorporated compared to earlier of inhomogeneities have been added (Appendix A2). The ACMANT versions: (1) Only the homogenized periods two test datasets differ only in the number and com- of partner series are taken into account other than at the pleteness of their time series, and in the spatial corre- early phase of the homogenisation procedure, or when lations between the test series. Each dataset consists of the interpolation is for a missing value out of the homog- 200 networks of 100-year long time series. In dataset enized period of the candidate series; (2) changes to the (A), the spatial correlations are around 0.9, and each net- parameterisation scheme. work includes 10 time series with a low missing data ratio. In dataset (B), the mean spatial correlation is 0.8, 3.10. Gap filling of daily data before homogenisation each network contains 15 time series, and the missing The advantage of gap filling on data at daily scale before data ratio is elevated. Table 2 summarizes the missing homogenisation is that temporally fragmented observed data characteristics of datasets (A) and (B) where miss- values can be incorporated into the homogenisation. The ing data due to the shortage of the period of observa- interpolation of daily values is performed in the same way tions are referred to external missing data, and those as that of the monthly values in monthly homogenisation, within the period of observations internal missing data, other than for some changes in the parameterisation respectively. scheme applied. This interpolation is repeated several It can be seen that although in dataset (B) networks times during a homogenisation procedure, as after the include 15 time series, the total number of observed val- adjustments for biases caused by inhomogeneities, more ues is lower with 16% in dataset (B) than in dataset (A). accurate interpolated values can be provided for infilling Homogenisation efficiency measures are presented here data gaps. via the comparison of the residual errors with the raw data errors. The examined efficiency measures are: (1) monthly 3.11. Completion of time series root mean squared error (RMSE), (2) annual RMSE, (3) absolute trend bias of individual time series, (4) absolute For ACMANT3, even if the input data comprises time network mean trend bias, (5) systematic network mean series of various lengths, the homogenized output series trend bias. Figures 1 and 2 present the raw data errors and are (optionally) completed with data for the same time the errors after the ACMANT homogenisation with three period. These are generally from the earliest year with ACMANT versions from the earliest to the most recent: available observational data in the network through until ACMANT1 (AC1), ACMANT2 (AC2) and ACMANT3 the latest year with observational data in the network. (AC3). Dataset (B) cannot be homogenized with AC1. In the other homogenisation results, three test statistics are 4. Efficiency tests shown for each method version and the associated effi- For measuring the efficiency of homogenisation methods, ciency measure: mean error, value of percentile 0.95 and artificially generated test datasets are needed, in which maximal error, other than for the systematic network mean the positions, shapes and magnitudes of inhomogeneities error where only the mean error is presented.

(a) (b)

Figure 1. Errors of raw data and those of ACMANT homogenisation products for dataset (A). (a) Monthly RMSE, (b) annual RMSE, (c) trend bias for individual series, (d) network mean trend bias. Smean means systematic trend bias.

The main characteristics of the results are as follows: values (with the exception of monthly RMSE). The difference between the two datasets is more marked in 1 The statistics are always better for the ACMANT relation to the maximal residual errors having occurred homogenisation results than for the raw data with in the homogenisation of the 200 networks. Consid- the exception of maximum monthly RMSE in dataset ering that in dataset (B) both the amount of observed (B). The largest reduction of raw data errors (∼75%) data and the spatial correlations are lower, and that the are associated with the individual trend bias and internal missing data ratio is higher than in dataset (A), systematic network mean trend bias in dataset (A). the efficiency of the homogenisation of dataset (B) with However, it is also worth noting that the improvement ACMANT2 or ACMANT3 is acceptable. is strongly related to the statistical characteristics of the 4 The comparison of residual errors between ACMANT inhomogeneities in the data. versions shows a slight but consistent improvement of 2 Although inhomogeneities are inserted into datasets efficiency in the transition from older versions to newer (A) and (B) according to the same rules, the raw versions based on each efficiency measure and the test data errors are slightly lower in dataset (B) than in statistic applied. Relatively large improvement can be dataset (A). This is because the mean length of peri- seen in the residual trend biases, particularly at the 0.95 ods with observed data is shorter in dataset (B) than in percentile and in the systematic network mean trend bias dataset (A). as well. 3 Residual homogenisation errors for dataset (B) are con- sistently higher than for those of dataset (A), but the dif- Figure 3 shows the accuracy of the interpolation for ferences are generally not very large; e.g. the increment missing monthly temperatures in datasets A and B in is smaller than 50% for the mean errors (except for the function of the number of the partner series used. If no systematic network mean bias) and the 0.95 percentile partner series could be used then the empirical climatic

(a) (b)

Figure 2. The same as Figure 1, but for dataset (B).

°C series and the spatial correlations between time series. 2 The RMSE of interpolated values is mostly smaller with 1.6 45–65% than the RMSE of the empirical climatic mean. Note that the accuracy of annual values is markedly 1.2 higher than that of the monthly values even when each monthly value of the year is interpolated. For such years, 0.8 the mean residual annual RMSE is 0.29 ∘C in dataset ∘ 0.4 A and 0.34 C in dataset B. These results indicate that the accuracy of annual values from interpolated monthly 0 values tends to be slightly higher than the accuracy of 012345678 9 the annual values from observed monthly values before A B homogenisation.

Figure 3. RMSE of interpolated monthly values of the homogenized datasets A and B. In the horizontal axis, the number of partner series used is shown. 5. Discussion and recommendations Approximately 20 years have passed since the multiple mean substitutes the missing data, thus, the other pieces break theory was proposed. The underpinning princi- of the results can be compared with the error of this ple of this theory is simple and is not limited to the simple substitution occurred in dataset B. The results topic of time series homogenisation: when a physical show that the accuracy depends on the number of partner problem includes mutually dependent factors, adequate

© 2016 Royal Meteorological Society Int. J. Climatol. 37: 1910–1921 (2017) HOMOGENISATION OF TIME SERIES WITH ACMANT3 1917 mathematical methods must include the joint treatment of preparation and the joint use of ACMANT3 with the soft- these factors. Although the frequency of the application ware ‘Rclimdex-extraQC’ for the common quality control of HOMER has increased exponentially in the recent of daily temperature and precipitation data. years (e.g. Freitas et al., 2013; Mamara et al., 2014; The use of ACMANT is recommended with the follow- Noone et al., 2016) due to the recommendation of HOME, ing caveats: the overall frequency of using multiple break methods is still low, and there are indications that the general 1. A general quality control of data before the applica- understanding of the multiple break homogenisation tion of ACMANT is necessary. Date order errors or concept is still missing. Although HOMER was the first accidental mixing of station series might cause serious recommended method of HOME, we do not yet have clear errors in the final results, and may even impede the evidence whether MASH, HOMER or ACMANT have execution of the programmes. Physical outliers due ∘ the highest efficiency when they are applied to particular to data transcription errors (e.g. 100 C instead of ∘ homogenisation tasks. Considering the similarities of 10.0 C) might result in serious biases in the calcu- the theoretical background of HOMER and ACMANT, lation of climatic means affecting the final results of it seems reasonable to assume that the efficiency of homogenisation. At current development, the inner these two methods is likely to be similar. Therefore, the quality control routine of ACMANT for filtering choice between HOMER and ACMANT for particular spatial outliers is not sufficient without the previous homogenisation tasks should be based on the dataset filtering of physically implausible values. Frequency characteristics rather than on the efficiency order (even if of zero precipitation events should be checked before this were known). The use of ACMANT is particularly using ACMANT, as zeros are sometimes erroneously recommended for (1) datasets with little or no metadata; shown instead of missing data code in climate (2) datasets from dense networks with large numbers of records. time series and where there are high spatial correlations; 2. Synchronous breaks might affect seriously the effi- (3) very large datasets (>∼200 time series) for which the ciency of homogenisation, as the concept of relative use of automatic methods is the most feasible and easily homogenisation is that breaks can be identified from managed solution. the differences of the candidate and reference time One important purpose of homogenisation is to deliver series. The potential danger is obvious when the syn- regional and global mean temperature trend estimates chronous break is present in at least in half of the which are more accurate (Rohde et al., 2013; Rennie et al., time series of a network and tests indicate (not shown) 2014; Venema et al., 2015). The efficiency tests presented that the efficiency declines even when the ratio of here provide firm indications that ACMANT3 can consid- affected time series is much lower than 50%. How- erably reduce initial regional trend biases at any spatial ever, synchronous breaks are often the consequences scale, although the efficiency achieved depends both on of new protocols in observing networks, and hence, the spatial density and the extent of the intact record of they are often well documented. It is recommended the observational data. Further research is needed in this to apply adjustments for known synchronous breaks important and emerging area, for both the development before using ACMANT, even if the precise break mag- and testing of statistical methods (Domonkos and Gui- nitudes are not known. jarro, 2015) and alongside an analysis of the causes of pos- 3. If a break magnitude is known from parallel measure- sible systematic biases in temperature records, with par- ments, then it is the best to apply the known correction allel measurements (http://www.surfacetemperatures.org/ term before using ACMANT. databank/parallel_measurements). 4. The use of raw climate records is preferred as the Certainly in the case of ACMANT, there is still room input for ACMANT rather than products of previ- for further development and refinement. For instance, the ous homogenisations or other kinds of secondary data ACMANT3 daily homogenisation programmes do not products. In the optimal case, the ACMANT input does yet contain varied adjustment terms according to the per- not contain interpolated values or adjustments based on centiles of the probability distribution function, while such spatial comparisons other than the adjustments applied procedures are included in some other homogenisation for synchronous breaks if those are reasonable. methods, e.g. Della-Marta and Wanner (2006); Kuglitsch 5. The maximum number of time series that can be et al. (2009); Mestre et al. (2011). The analysis of the homogenized within one network is 99 using the frequency changes of the 0 values in precipitation time ACMANT homogenisation methods, and the use of series can also be incorporated in automatic homogenisa- very large networks is not recommended. The opti- tion as it is shown by Wang et al. (2010). Nevertheless it mal size of networks is usually around 20–30 time is considered that the main phases in the development of series, although when the potential reference series ACMANT as a fully automatic homogenisation method are shorter than the candidate series or where they for temperature and precipitation homogenisation have are often incomplete, then the optimum number of been completed. time series can be much higher. For large and dense Beyond the methodological improvements, ACMANT3 networks, it is recommended to apply automatic is also more user-friendly. Of particular note, the soft- networking following the suggestions of Domonkos ware package includes auxiliary files supporting input data (2015b).

The ACMANT3 software package together with its man- ∑N g ual is freely accessible from http://www.c3.urv.cat/data Fg h ,h = wgA , (A1) [ 1 2] s[h1 h2] .html. s=1 The weights of the reference composites are deter- Acknowledgements mined by ordinary kriging (Szentimrey, 2010) if N > 5, The research was funded by the Spanish project while by the spatial correlations with the candidate series ‘Multiple verification of automatic softwares homog- (Domonkos, 2011b) in the reverse case. enizing monthly temperature and precipitation series’ Break detection and outlier detection are performed on CGL2014-52901-P. JC acknowledge funding provided by relative time series (Q). the Irish Environmental Protection Agency under project Q = A , − F , (A2) 2012-CCRP-FS.11. g g[h1 h2] g[h1 h2]

Appendix A1: Mathematical formulations A1.1. Explanation of symbols A1.3. Determination of the number of breaks and step function fitting with predetermined number of steps A – time series of deseasonalized observed values B – preliminary adjustment term Optimal step function fitting with K steps: The task of optimal step function fitting as a model of c, cp – parameter d – internal distance time series is identical with the minimisation of the vari- e – external distance ance of internal distances (variation of observational data F – composite reference series within constant sections of the model) and maximisation g – index of candidate series of the variance of external distances (variation of the mod- h – length of period elled values, Lindau and Venema, 2013). The step function h′ – number of available data in a period has K + 1 constant sections (k = 0, 1, 2, … K). h1, h2 – starting and ending points of period in any time Internal distance (d): scale d (U)i = U (q)i − U (q)k where i ∈ k (A3) i, i* – time point External distance (e): j – year e (U)i = U (q)k − U (Q) where i ∈ k (A4) j1, j2 – starting and ending years U is operator, most frequently (but not always) the k – serial number of break/step generator of time average. K – total number of breaks (1) Univariate detection m – calendar month { } n – number of years in time series ∑K ∑hk+1 ( ) 2 N – total number of stations in network or of reference min d (U) , (A5) , , k i [h1 h2 … hK ] series k=0 i=hk+1 ′ N – total number of usable reference series at a partic- (2) Bivariate detection ular step P – penalty term { ( )} ∑K ∑jk+1 ( ) ( ( ) ) 2 2 Q – relative time series min d U + c d U , , 1 k,i 2 k,i r – spatial correlation [j1 j2 … jK ] k=0 i=jk+1 s – serial number of reference series (A6) T – matrix The bivariate detection is always performed on series U – operator with annual resolution. c is empirical constant (c = 0.2) in w – weight temperature homogenisation, while it is the squared ratio W – accumulated weight of snowy months in proportion to the rainy months in x – serial number of ensemble homogenisation precipitation homogenisation. ′ Z, Z – adjustment term For univariate detection on annual scale and for bivariate upper stroke – section mean or mean of entire time detections: series ≥ ≤ ≤ jk+1 − jk j ∗ for every k ∈ {0 k K} (A7) A1.2. Reference series and relative time series In most operations j* = 3, but in the break detection for Reference series (F) are built for candidate series (G) annual temperature series of particular calendar months are composed from the other time series (As)ofthe j* = 5. same climatic network. The reference series often covers In determining the optimal number of steps, the modi- h h only a section [ 1, 2] of the candidate series. Reference fied Caussinus–Lyazrhi criterion (C–L) is used, in which series usually contain entire years, but the reference series expression (A8) is minimized. This criterion is used only with daily resolution for operations on daily scale are the in annual scale detections. exceptions.

⎧ ∑K ( ) [( ( ) ) ( ( ) ) ] ⎫ ⎪ 2 2 ⎪ jk+1 − jk · e U1 k + c e U2 k ⎪ k=0 ⎪ ln ⎨1 − ⎬ + P (A8) h {( ) ( ) } ⎪ ∑ 2 2 ⎪ ⎪ U (q) − U (Q) + c U (q) − U (Q) ⎪ ⎩ 1 i 1 2 i 2 ⎭ i=1

2K P = c ln (h) (A9) p h − 1 Note (1) that (A8) and (A9) differ only by the inclusion years (i.e. they start with 01 January and end with 31 of coefficient cp in the penalty term from the original December). The weights are set subjectively, and they C–L (Caussinus and Lyazrhi, 1997); (2) When c = 0, the depend on factors influencing the usability of values in formula is usable for univariate detection. partner series in contributing to the accurate estimation of Usually cp = 1.4 in univariate detection and cp = 1.0 in the target value: (1) spatial correlation (rg,s); (2) data avail- bivariate detection, but in the first pre-homogenisation the ability around i*; (3) whether data are pre-homogenized coefficient is elevated with 40%. or not; (4) the phase of the homogenisation (as the interpolation is repeatedly performed in ACMANT, with data A1.4. Taking the minimum correction terms of of increasing quality); (5) daily or monthly interpolation is ensemble homogenisation applied. It follows from (A12) and (A13) that if no data of partner Phase 1: Let us assume we have N time series in a given series is available, then the missing data of the candidate network where the length of the studied period is n years. series will be substituted with the climatic mean. For any time series s, the possible highest number of ensemble members is N – 1, as it is the maximum possi- A1.6. Smoothing of monthly adjustment terms ble number of reference components. Due to the different lengths of time series or low spatial correlations the real The connection between the preliminary adjustment terms numbers of ensemble members (N′) for a particular sta- (b) and final adjustment terms (z′) is shown by (A14). tion and particular year can be lower, so that N′ ≤ N − 1 s,j ′ , ,.. for any s and j. For each ensemble member, the match- zj,m = 0.3bj,m−1 + 0.4bj,m + 0.3bj,m+1 m = {1 2 .12} ing adjustment terms (T) are calculated with ANOVA on (A14) annual scale and these preliminary results are stored. T is a three dimensional matrix with the dimensions of station bj,0 = bj−1,12; bj,13 = bj+1,1 (A15) serial number (s), year (j), and the serial number of ensemble experiment (x). Phase 2: The final adjustment in a pre-homogenisation Appendix A2: Additional noise, climatic trends and procedure (Z) for any station s and year j is: inhomogeneities in test datasets (A) and (B). | | ( ) ⎧ | | ⎫ Additional noise is applied only to dataset (B), while the ⎪ {min } |ts,j,x| if sign ts,j,x = c for every x⎪ x= 1,N′ other operations are applied uniformly to datasets (A) and z = ⎨ s,j { } ⎬ s,j ( ) (B). ⎪ ≠ , ′ ⎪ ⎩ 0ifsign ts,j,x c for any x ∈ 1 Ns,j ⎭ (A10) A2.1. Definitions

c = 1 or c =−1 (A11) 1. Central series: The series which has the highest spatial A1.5. Interpolation for substituting missing values correlations on average with the other series of the same network, is named the central series. 2. Limit bias: Biases of multiple inhomogeneities of the ∑N j∑2(g) 1 2 1 same time series may be accumulated until the parame- ag,i∗ = c (h) rg,sas,h(s)[j1,j2] + ag,i (A12) W h′ ter of subjectively defined limit values, they are named s=1 i=j1(g) { } limit biases. ∑N 3. Platform: Platform-shaped inhomogeneity, pair of , 2 W = max 0.4 c (h) rg,s (A13) shifts of the same sign and opposite directions. s=1 The lengths of the periods (h) and the number of data A2.2. Additional noise ′ utilized (h ) for the calculation of empirical climatic mean A monthly series of normally distributed red noise with around the date for which the interpolation is performed 0 mean and 0.15 autocorrelation is added. The variance (i*) depend on the data availability in the partner series. is a function of the spatial correlation with the central However, periods taken into account always include entire series, and the parameters of this function are empirically

© 2016 Royal Meteorological Society Int. J. Climatol. 37: 1910–1921 (2017) 1920 P. DOMONKOS AND J. COLL determined for obtaining the target mean spatial correla- 7 Seasonal cycle of biases: 25% of the inhomogeneities tioninnetworks. are without seasonal variation of bias. In the other 75%, the form of the variation is semi-sinusoid with A2.3. Additional climatic trends modes in July and December (so that the spring half period lasts 7 months, while the autumn half period Each network receives one additional climatic trend. The lasts 5 months). The magnitude is a random variable of trend magnitude is a random variable of uniform distribu- ∘ uniform distribution. The peak-to-peak amplitude (i.e. tion between 0 and 2 C. 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